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Properties

Label 30576.bi
Number of curves $6$
Conductor $30576$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -138692752, -628632494048]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -138692752, -628632494048]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -138692752, -628632494048]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 30576.bi have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 30576.bi do not have complex multiplication.

Modular form 30576.2.a.bi

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{3} + 2 q^{5} + q^{9} + 4 q^{11} - q^{13} - 2 q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 30576.bi

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
30576.bi1 30576d6 \([0, -1, 0, -138692752, -628632494048]\) \(1224522642327678150914/66339\) \(15984060438528\) \([2]\) \(1769472\) \(2.9242\)  
30576.bi2 30576d4 \([0, -1, 0, -8668312, -9820179200]\) \(597914615076708388/4400862921\) \(530183292715754496\) \([2, 2]\) \(884736\) \(2.5777\)  
30576.bi3 30576d5 \([0, -1, 0, -8489952, -10243819872]\) \(-280880296871140514/25701087819771\) \(-6192552511300072200192\) \([2]\) \(1769472\) \(2.9242\)  
30576.bi4 30576d3 \([0, -1, 0, -1849472, 798272112]\) \(5807363790481348/1079211743883\) \(130015418835037252608\) \([2]\) \(884736\) \(2.5777\)  
30576.bi5 30576d2 \([0, -1, 0, -552932, -146646240]\) \(620742479063632/49991146569\) \(1505640551090247936\) \([2, 2]\) \(442368\) \(2.2311\)  
30576.bi6 30576d1 \([0, -1, 0, 35313, -10408698]\) \(2587063175168/26304786963\) \(-49515710102559792\) \([2]\) \(221184\) \(1.8845\) \(\Gamma_0(N)\)-optimal